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liyz06
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Homework Statement
How should I expand 1/(1+x)^n around x=0?
Homework Equations
The Attempt at a Solution
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mfb said:Did you try a Taylor expansion?
What did you get? Where did you run into problems?
Are you familiar with Taylor Series ?liyz06 said:Homework Statement
How should I expand 1/(1+x)^n around x=0?
Homework Equations
The Attempt at a Solution
The general formula for expanding 1/(1+x)^n around x=0 is (1+x)^(-n) = 1 - nx + (n(n+1)/2!) x^2 - (n(n+1)(n+2)/3!) x^3 + ... This is known as the Binomial Series expansion.
It is recommended to stop expanding when the term involving x^k is smaller than the desired level of accuracy. This means that we stop expanding when |(n(n+1)...(n+k-1)/k!) x^k| < ε, where ε is the desired level of accuracy.
The expansion for 1/(1+x)^n around x=0 is valid for any real value of n. However, it may not converge for values of n that are negative or non-integer.
Expanding 1/(1+x)^n around x=0 allows us to approximate the value of the function for values of x near 0. This can be helpful in simplifying complex expressions and solving differential equations.
Yes, there are other methods such as using the Newton's Binomial Theorem or using the Taylor Series expansion for 1/(1+x). However, the Binomial Series expansion is the most commonly used method for expanding 1/(1+x)^n around x=0.